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Unformatted text preview: Lecture 17  4/7/2010 Discrete Stochastic Processes 1 Discrete Stochastic Processes Lecture 17 Midterm Quiz Discussion of a few problems CountableState Markov Chains Branching Processes – Section 5.5 Lecture 17  4/7/2010 Discrete Stochastic Processes 2 Problem 1 (19 pts.) Consider the following finitestate Markov Chain. (5) a) Identify all the classes present in the chain and the states belonging to each class. Find the period of each class and determine whether the class is transient or recurrent. b) Let ( ) , i j p n denote the probability of the process ending up in state j in n transitions, conditioned on the fact that the process started in state i . In other words, ( ) ( ) , i j n p n P X j X i = = = . Compute the value of each of the limits below, or else explain briefly why it does not exist. (2) i) ( ) 1,5 lim n p n →∞ . (2) ii) ( ) 1,7 lim n p n →∞ . (2) iii) ( ) 1,2 lim n p n →∞ . (2) iv) ( ) 4,5 lim n p n →∞ . (6) c) Let , i j P p ⎡ ⎤ = ⎣ ⎦ be the transition matrix for this chain. Find all the possible steady state vectors for this chain, i.e., find all vectors [ ] 1 2 7 , , , π π π π = K with the properties that 1 2 7 1 7 1, 0 , , 1 π π π π π + + + = ≤ ≤ K K and P π π = . Lecture 17  4/7/2010 Discrete Stochastic Processes 3 Problem 2 (20 pts) Consider a car ferry that holds some integer number k of cars and carries them across a river. The ferry business has been good, but customers complain about the long wait for the ferry to fill up. (5) a) Assume that cars arrive according to a renewal process. The IID inter arrival times have mean X , variance 2 σ and moment generating function ( ) X g r . The ferry departs immediately upon the arrival of the k th customer and subsequent ferries leave immediately upon the arrival of the 2 k th customer, the 3 k th customer, etc. Does the sequence of departure times of the ferries form a renewal process? Explain carefully. (5) b) Find the expected time that a randomly chosen customer waits from arriving at the ferry terminal until departure of its ferry. As part of your solution, please give a reasonable definition of the expected waiting time for a randomly chosen customer, and please first solve this problem explicitly for the cases k = 1 and k = 2. (5) c) Is there a 'slow truck' phenomenon here ? (This the phrase we used to describe the effect of large variance on the term E[X 2 ]/2E[X] in the steady state residual life or on the E[Z 2 ] term in the numerator of the PollazcekKhinchin formula.) Give a brief intuitive explanation. (5) d) In an effort to decrease waiting, the ferry managers institute a policy where the maximum interval between ferry departures is 1 hour. Thus a ferry leaves either when it is full or after one hour has elapsed, whichever comes first. Does the sequence of departure times of ferries that leave with a full load of cars constitute a renewal process? Explain carefully. Lecture 17  4/7/2010...
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This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.
 Spring '08
 Moon,J

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